Canonical Expansion Research Articles

Simultaneous statistical modeling of monthly precipitation amount at several observational points

Demonstrated is the possibility of the group modeling of the series of monthly precipitation amount using the method of canonical expansion. Used are the algorithms and software computing programs developed as applied to the statistical models of the monthly and annual river runoff. The model of canonical expansion preserves all statistical characteristics of initial series.

Towards uniformly $\Gamma$-equivalent theories for nonconvex discrete systems

In this paper we consider a one-dimensional chain of atoms whichinteract with their nearest and next-to-nearest neighboursvia a Lennard-Jones type potential. We prescribe the positionsin the deformed configuration of the first two and the last two atomsof the chain.  &nbsp We are interested in a good approximation of the discrete energy of thissystem for a large number of atoms, i.e., in the continuum limit.  &nbsp We show that the canonical expansion by $\Gamma$-convergence does not provide an accurate approximationof the discrete energy if the boundary conditions for the deformation are close to the threshold betweenelastic and fracture regimes.This suggests that a uniformly $\Gamma$-equivalent approximation of the energy should be made, asintroduced by Braides and Truskinovsky, to overcome the drawback of the lack of accuracy of the standard$\Gamma$-expansion.  &nbsp In this spirit we provide a uniformly $\Gamma$-equivalent approximation of the discrete energy at first order, which arisesas the $\Gamma$-limit of a suitably scaled functional.

Infinitesimal affine transformations of the second order tangent bundle with horizontal lift connection

We obtain the canonical expansion of an infinitesimal affine transformation of the second order tangent bundle with horizontal lift connection. We establish necessary and sufficient conditions under which a vector field is an infinitesimal affine transformation. We also construct the horizontal lift of a linear connection to a second order Weil bundle.

Cartan–Laptev method in the theory of multidimensional three-webs

We show how the Cartan–Laptev method that generalizes Elie Cartan’s method of external forms and moving frames is applied to the study of closed G-structures defined by multidimensional three-webs formed on a C s-smooth manifold of dimension 2r, r ≥ 1, s ≥ 3, by a triple of foliations of codimension r. We say that a tensor T belonging to a differential-geometric object of order s of a three-web W is closed if it can be expressed in terms of components of objects of lower order s. We find all closed tensors of a three-web and the geometric sense of one of relations connecting three-web tensors. We also point out some sufficient conditions for the web to have a closed G-structure. It follows from our results that the G-structure associated with a hexagonal three-web W is a closed G-structure of class 4. It is proved that basic tensors of a three-web W belonging to a differential-geometric object of order s of the web can be expressed in terms of an s-jet of the canonical expansion of its coordinate loop, and conversely. This implies that the canonical expansion of every coordinate loop of a three-web W with closed G-structure of class s is completely defined by an s-jet of this expansion. We also consider webs with one-digit identities of kth order in their coordinate loops and find the conditions for these webs to have the closed G-structure.

Functional Singular Component Analysis

Summary Aiming at quantifying the dependence of pairs of functional data (X,Y), we develop the concept of functional singular value decomposition for covariance and functional singular component analysis, building on the concept of ‘canonical expansion’ of compact operators in functional analysis. We demonstrate the estimation of the resulting singular values, functions and components for the practically relevant case of sparse and noise-contaminated longitudinal data and provide asymptotic consistency results. Expanding bivariate functional data into singular functions emerges as a natural extension of the popular functional principal component analysis for single processes to the case of paired processes. A natural application of the functional singular value decomposition is a measure of functional correlation. Owing to the involvement of an inverse operation, most previously considered functional correlation measures are plagued by numerical instabilities and strong sensitivity to the choice of smoothing parameters. These problems are exacerbated for the case of sparse longitudinal data, on which we focus. The functional correlation measure that is derived from the functional singular value decomposition behaves well with respect to numerical stability and statistical error, as we demonstrate in a simulation study. Practical feasibility for applications to longitudinal data is illustrated with examples from a study on aging and on-line auctions.

Effective valence-shell hamiltonians generated by canonical transformations

The salient points of applying the canonical transformation-cluster expansion formalism to the problem of generating effective valence-shell Hamiltonians are reviewed. Substantial reduction in the strength of the two-body interaction is found to be in agreement with the semiempirical expectations. The low-lying excitation spectrum of the carbon atom is calculated by diagonalizing the effective valence-shell Hamiltonian, and the results agree well with experiment. Use of the formalism for obtaining from first principles the effective one-and two-body interactions to be used in very large molecular structure calculations is proposed.

15 years of very long baseline interferometry observations of two compact radio sources in Messier 82

We present the results of a second epoch of 18-cm global very long baseline interferometry (VLBI) observations, taken on 2001 February 23, of the central kiloparsec of the nearby starburst galaxy Messier 82. These observations further investigate the structural and flux evolution of the most compact radio sources in the central region of M82. The two most compact radio objects in M82 have been investigated (41.95+575 and 43.31+592). Using this recent epoch of data in comparison with our previous global VLBI observations and two earlier epochs of European VLBI network observations, we measure expansion velocities in the range of 1500–2000 km s−1 for 41.95+575 and 9000–11 000 km s−1 for 43.31+592 using various independent methods. In each case, the measured remnant expansion velocities are significantly larger than the canonical expansion velocity (500 km s−1) of supernova remnants (SNRs) within M82 predicted from theoretical models. In this paper, we discuss the implications of these measured expansion velocities with respect to the high-density environment that the SNRs are expected to reside in within the centre of the M82 starburst.

A Modification of the Canonical Grid Series Expansion in Order to Increase the Efficiency of the SMCG Method

In this letter, we present a Green's function approximation valid in the weak-interaction region that can be used with the sparse-matrix canonical grid (SMCG) method. It can be easily introduced into existing SMCG codes, allowing a reduction in size of the neighborhood region and, consequently, of the dynamic memory and the computation time requirements.

Bi-framelet systems with few vanishing moments characterize Besov spaces

We study the approximation properties of wavelet bi-frame systems in L p( R d) . For wavelet bi-frame systems the approximation spaces associated with best m-term approximation are completely characterized for a certain range of smoothness parameters limited by the number of vanishing moments of the generators of the dual frame. The approximation spaces turn out to be essentially Besov spaces, just as in the classical orthonormal wavelet case. We also prove that for smooth functions, the canonical expansion in the wavelet bi-frame system is sparse and one can reach the optimal rate of approximation by simply thresholding the canonical expansion. For twice oversampled MRA based wavelet frames, a characterization of the associated approximation space is obtained without any restrictions given by the number of vanishing moments, but at a price of replacing the canonical expansion by another linear expansion.

Canonical expansion of -symmetric operators and perturbation theory

Let $H$ be any $\PT$ symmetric Schr\"odinger operator of the type $ -\hbar^2\Delta+(x_1^2+...+x_d^2)+igW(x_1,...,x_d)$ on $L^2(\R^d)$, where $W$ is any odd homogeneous polynomial and $g\in\R$. It is proved that $\P H$ is self-adjoint and that its eigenvalues coincide (up to a sign) with the singular values of $H$, i.e. the eigenvalues of $\sqrt{H^\ast H}$. Moreover we explicitly construct the canonical expansion of $H$ and determine the singular values $\mu_j$ of $H$ through the Borel summability of their divergent perturbation theory. The singular values yield estimates of the location of the eigenvalues $\l_j$ of $H$ by Weyl's inequalities.

On the nature of the contributions to the entropy of Pauling ice

A calculation is presented for the entropy and correlation functions of Pauling ice—a model in which the energy of the microstates can take only two values: zero or infinity. The central point of the proposed approach is the use of a canonical expansion of the thermodynamic functions in irreducible multiparticle correlations. Reduction rules are formulated which establish the relation between the correlation functions of orders 1, …, k−1 and the kth-order correlation function of the system. The latter is calculated on the basis of the assumption of equal probabilities of all allowed configurations of a compact group of k particles and the normalization condition. The order of approximation is determined by the number of particles k for which the correlations between states is taken into account. It is shown that with increasing k the values of the entropy of two-dimensional “square ice” converge nonmonotonically to its exact value obtained by Lieb. The best agreement corresponds to those approximations in which the group of k particles has the lattice symmetry and contains closed loops of hydrogen bonds. The method can be extended to arbitrary lattice systems.

Characterization of vegetation in the south of M@xico by means of a canonical expansion

In this work, a method to characterize vegetation cover types is presented. The state of vegetation is represented by means of canonical variables derived from the bands of a multi-spectral image. The canonical variables handled in this work are the following: albedo of vegetation, strength of the greenness of vegetation, and humidity content of vegetation. These variables are uncorrelated, as is demonstrated in this research. The ensemble of these variables establishes a canonical expansion of the multi-spectral image. This is a canonical representation of vegetation. The canonical variables mentioned above form a multi-band image. This image is input into an unsupervised spectral classifier to generate a thematic map of vegetation cover types. Based on ground data, these vegetation classes are identified. In addition to this map, the following information layers are incorporated into the characterization process: lithological units, digital terrain model, hydrography, geomorphological units and climate. The whole of this information is integrated into a geospatial database to provide a description of the vegetation classes.

Paired-permanent approach for VB theory (II)

Paired-permanent approach for VB theory is extensively developed. Canonical expansion of a paired-permanent is deduced. Furthermore, it is shown that a paired-permanent may be expressed in terms of the products of sub-paired-permanents of any given order and their corresponding minors. An ab initio spin-free valence bond program, called Xiamen, is implemented by using paired-permanent approach. Test calculation shows that Xiamen package is more efficient than some other programs based on the traditional VB algorithm, and it provides a new practical tool for quantum chemistry.

COSMOLOGICAL CONSTANT OF THE INFLATING BRANES IN THE RANDALL–SUNDRUM–ODA MODEL

We examine the issue of the cosmological constant in the many inflating branes scenario, extending on two recent models by I. Oda and Randall–Sundrum. The exact solution in a closed form is found in the slow-roll approximation for the branes. We show that each positive tension brane has an induced canonical cosmological constant and expansion rate that are decreasing functions of their position in the fifth dimension. The reason is that the square of the effective expansion rate enters as a source term in the Einstein equations for the branes. Thus the brane has two scale factors depending on time and the fifth dimension respectively. A displacement of the brane along the fifth dimension would adjust its effective expansion rate in such a way that it compensates for its internal energy changes due to inflation and possible phase transitions.

P-ADIC DYNAMICAL SYSTEMS AND NEURAL NETWORKS

A p-adic model which describes a large class of neural networks is presented. In this model the states of neurons are described by digits in the canonical expansion of a p-adic number. Thus each p-adic number represents a configuration of firing and non firing neurons. The process of recognition of patterns is investigated in the P-adic framework. We study heteroassociative and autoassociative nets. P-adic dynamical systems are used to describe a feedback process for autoassociative nets.

Modulo correlativity and its application in a multiple valued logic system

Several new concepts, such as pseudoprime and modulo correlativity, are presented. Then modulo subtraction and modulo division, which are very useful in computing the canonical expansion, are proposed in a multiple valued modulo algebra system with either a prime or a composite radix. Finally the completeness of modulo operations and the current mode CMOS circuit design for any unary function are discussed on the basis of modulo correlativity. This method is also applicable to a high-radix multiple valued modulo algebra system.

On the canonical grid method for two-dimensional scattering problems

The banded matrix iterative approach with a canonical grid expansion (BMIA/CAG) has been shown by Tsang et al. (see ibid., vol.43, p.851-9, 1995) to be an efficient method for the calculation of scattering for near planar two-dimensional (2-D) geometries such as one-dimensional rough surfaces. However, in the article of Tsang et. al, only the first three terms in the canonical grid series for TE polarization above a perfectly conducting surface were discussed and methods for implementing only a portion of these terms were presented. In this paper, a general form for all terms in the canonical grid series is provided for both TE and TM polarizations above an impedance surface and an efficient algorithm for calculating their contributions is described. The relationship between the canonical grid and operator expansion methods is also discussed. A sample surface scattering problem is shown to illustrate the utility of higher order terms in the canonical grid method.

A numerical study of low-grazing-angle backscatter from ocean-like impedance surfaces with the canonical grid method

A numerical study of 14-GHz low-grazing-angle (LGA) backscattering from ocean-like surfaces described by a Pierson-Moskowitz spectrum is presented. Surfaces rough in one dimension are investigated with Monte Carlo simulations performed efficiently through use of the canonical grid expansion in an iterative method of moments. Backscattering cross sections are illustrated at angles from 81/spl deg/ to 89/spl deg/ from normal incidence under the impedance boundary condition (IBC) approximation with the efficiency of the numerical model enabling sufficiently large profiles (8192 /spl lambda/) to be considered so that angular resolution problems can be avoided. Variations with surface spectrum low-frequency cutoff (ranging over spatial lengths from 175.5 m to 4.29 cm) at 3 m/s wind speed are investigated and initial assessments of the small perturbation method (SPM), composite surface theory, operator expansion method (OEM), small slope approximation (SSA), and curvature corrected SPM predictions are performed. Numerical results show an increase in horizontal (HH) backscatter returns as surface low-frequency content is increased while vertical (VV) returns remain relatively constant, as expected, but none of the approximate models considered are found to produce accurate predictions for the entire range of grazing angles. For the cases considered, HH scattering is always observed to be below VV, further demonstrating the importance of improved hydrodynamical models if "super-event" phenomena are to be modeled.

Numerical method for investigation of stability of stochastic integro-differential equations

The present paper is devoted to the numerical solution of the stability problem for linear integro-differential equations, describing the behavior of stochastic viscoelastic systems. The method is based on the statistical simulation of input random wide-band stationary processes and with this aim the method of the canonical expansion is used. For each realization of the process, the numerical solution of integro-differential equations is found. With the help of Liapunov exponents, which are calculated for statistical moments of searched unknowns, the boundaries of the stability with respect to statistical moments of the required order can be obtained. The proposed method allows to investigate the stability of systems excited with stationary processes having the correlation function as well as probability distribution of wide applicability.

Matrix method for solving multivalued logic differential equations

A method to solve logic Differential equations, i.e. equations containing Logic Derivatives of k-valued logic (MVL) functions is proposed. An initial Differential equation is represented by a system of k logic equations of k variables given as 0-polarity Reed-Muller canonical expansion. This system is solved by means of a truncated orthogonal transform algorithm.